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Armstrong on Quantities and Resemblance Philosophical Studies (2007) 136: 385-404

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Armstrong on Quantities and Resemblance Philosophical Studies (2007) 136: 385-404 M. Eddon Armstrong on Quantities and Resemblance Philosophical Studies (2007) 136: 385-404 Abstract: Resemblances obtain not only between objects but between properties. Resemblances of the latter sort – in particular, resemblances between quantitative properties – prove to be the downfall of David Armstrong’s well-known theory of universals. This paper examines Armstrong’s efforts to account for such resemblances, and explores several ways one might extend the theory in order to account for quantity. I argue that none succeed. A theory of universals takes at face value the idea that things share properties. Such a theory holds that universals can be instantiated by numerically distinct objects. One of the natural applications of this theory is to explain how two things resemble one another, and thus to offer an answer to the so-called Problem of Resemblance: two things intrinsically resemble one another if and only if they share some of their universals.1 David Armstrong claims that universals provide the only tenable account of resemblance, because they provide the only reductive account (see Armstrong).2 But whether universals provide an attractive analysis of resemblance hinges on a crucial question: can a theory of universals account for resemblance relations among properties as well as resemblance relations among objects? Armstrong believes so. He offers an account according to which the more parts two properties share, the more similar they are.3 This strategy is fatally flawed. As a result, I argue, a theory of universals cannot count an analysis of resemblance among its virtues. Since one of its alleged strengths is an elegant and reductive analysis of resemblance, the failure to produce such an account is a mark against the theory. (I will not be weighing other costs and benefits here.) In this paper I will look at how Armstrong’s theory deals with quantitative properties, particularly those of classical mechanics. I do this for three reasons. First, Armstrong himself claims that universals are in a unique position to accommodate quantitative properties.4 Second, a world where the laws of classical mechanics hold is metaphysically possible, and Armstrong should be able to 1 In this paper I am interested only in intrinsic resemblance, not extrinsic resemblance. For instance, I do not address cases where there is some sense in which two things resemble (perhaps each has the property of being five feet from a poodle), but where this resemblance does not arise from the intrinsic properties of each object alone. 2 See Armstrong (1978) and (1989a). 3 See Armstrong (1988) and (1989a, pg. 101-105). 4 See Armstrong (1989a, pg. 101). account for such a world. Third, if Armstrong’s theory cannot accommodate the properties of classical mechanics, there is little hope it will be able to accommodate the quantitative properties of more sophisticated physical theories. I. Armstrong’s Picture: Resemblance as Partial Identity On Armstrong’s picture, universals are sparse; they carve nature at the joints. The paradigmatic universals are the fundamental quantities expressed by predicates in an ideal physics.5 Gruesome predicates have no correlates in the world of universals. Armstrong is especially conservative with his ontology of higher-order universals, or universals instantiated by universals. (The relation of nomic necessitation is one of the few higher-order relations he allows.) Although positing higher-order universals may seem a natural way to account for property resemblance, Armstrong has several reasons to reject this approach.6 As a result, he instead proposes a different strategy to account for resemblances among properties. Armstrong claims that universals can be constituents of other universals, just as objects can be parts of other objects. Universals made up of constituents are structural universals, while universals with no constituents are simple universals. On his account, the structure of universals mirrors the structure of the objects that instantiate them. Any object that instantiates a structural universal must have proper parts which instantiate that universal’s constituents. Consider a structural universal F with constituents F1 through Fn. If object a instantiates F, then a must have numerically distinct proper parts a1 through an which instantiate 7 F1 through Fn, respectively. Two structural universals resemble one another to the extent to which they share constituents. If two universals do not share any constituents, they do not resemble one another in any respect. If they share at least one constituent, they 5 See Armstrong (1988, pg. 87). 6 One reason is the desire for ontological parsimony. Another is the fact that the natural candidates for such higher-order universals are instantiated necessarily by first-order universals, which is at odds with Armstrong’s combinatorial view of possibility. See Armstrong (1978, pg. 105-108), (1983), and (1989b) for more discussion. 7 If one axiomatizes the part-whole relation using a mereology that abandons the axiom of unique fusion, then Armstrong’s constituency relation can be identified with the mereological relation of part to whole. On the other hand, if the part-whole relation is constitutively tied to the unique fusion axiom of classical mereology, then Armstrong must accept the constituency relation as a primitive in his ontology, albeit one that closely approximates the notion of traditional parthood. See Lewis (1986) for discussion. 2 resemble one another at least somewhat. The more constituents they share, the more similar they are: all resemblance is reduced to partial or whole identity. Armstrong claims that every quantitative property is a structural universal. Think of quantitative universals as Russian nesting dolls. Within each doll there is a smaller doll, and a smaller one, ad infinitum. The largest doll “contains” all the other dolls: it shares many nested dolls with the second-largest doll, slightly fewer nested dolls with the third-largest doll, and so on. In terms of the quantity of shared dolls, the largest is more similar to the second-largest than to the third- largest. Likewise for quantitative universals – every quantitative property has an infinite number of “nested” constituent universals. Intuitively, the more constituents two quantitative universals share, the more similar they are. For example, the five grams mass universal shares many constituents with the four grams mass universal; hence the property of five grams mass closely resembles the property of four grams mass. Armstrong’s remarks can be usefully formalized by the following two principles. The first principle provides an intuitively plausible way to determine the constituents of a structural universal. Call it the constituency principle: a universal x is a constituent of universal y iff every object in every possible world that instantiates y has some proper part that instantiates x. This principle links the structure of universals to the structure of objects – universals have constituents when the objects that instantiate them have parts. The second principle provides an intuitively plausible connection between the constituency relation and the resemblance relation. Let “x < y” mean “y has all of the constituents of x but x does not have all of the constituents of y,” where x and y are universals. Call this the resemblance principle: a is more similar to b than to c, and c is more similar to b than to a, iff a < b < c. This principle links resemblance to constituency – two properties are similar when they share constituents. Note that the resemblance principle applies only in cases where a, b, and c share at least one constituent. If they have no constituents in common, they are utterly dissimilar and so cannot be compared along any axis of similarity.8 The constituency principle does a tremendous amount of work for Armstrong. First, it provides an algorithm for determining the constituents of structural universals. It explains, for example, why a charge universal is never a constituent of a mass universal – because not every massive object has a charged proper part. It also explains why a mass universal never has constituents of greater 8 For supporting texts, see especially Armstrong (1978, pg. 116-131), (1988, pg. 312-316), and (1989a, pg. 106-107). 3 mass – because no object with mass x ever has a proper part with a mass greater than x. Second, the constituency principle grounds the structure of quantitative universals in the structure of objects and their proper parts. In so doing, it provides Armstrong with a justification for constructing quantitative universals as he does. Without this principle, we must brutely posit what the constituents of quantitative universals are. Tying resemblance to constituents that are themselves ungrounded is not an improvement over positing primitive resemblance. Thus, Armstrong analyzes resemblance in two steps. First, the constituency principle grounds the ontology of universals in the ontology of objects; second, the resemblance principle uses the ontology of universals to ground resemblance relations. In the following sections, I present several problems with this analysis. I conclude that Armstrong’s account is not a plausible theory of quantitative properties, and thus is not an improvement over accounts that posit primitive resemblance. II. The Metric Function Armstrong uses mass as a paradigm example to illustrate his account of quantities, so I will focus on it in the next two sections. However, my criticisms in these
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